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Relationship of E = mc2 to F = ma and Gravity
Copyright © 2016
Joseph A. Rybczyk
Abstract
A required modification of Newton’s second law of mechanics, F = ma force equation, to
make it compatible with Einstein’s E = mc2 energy equation is presented in the simplest terms
practical. An alternative explanation is then given for the real meaning of the modification that
is contrary to the mass increase explanation given in Einstein’s theory. In the process it is shown
that Newton’s second law of mechanics was not complete in its original form and that Einstein’s
Energy equation has nothing to do with the abstract principles of time dilation and distance
contraction of special relativity. Einstein’s energy equation is shown to involve nothing more
than Newtonian physics corrected for an omission in Newton’s treatment of force.
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Relationship of E = mc2 to F = ma and Gravity
Copyright Work © 2016 Joseph A. Rybczyk
1. Introduction
When all of the different factors are considered it becomes clear that the relationship
between Einstein’s energy equation1 and Newton’s inertial force equation is quite simple and
straight forward. The problem is that there are so many factors involved that it is easy to miss
the most obvious relationships in the total combined relationships that surface during the
investigative process. This appears to be the case involving the author’s previous research
documented in a paper2 published in June 2007. Although the paper is very enlightening it can
be overpowering in its depth of coverage and thereby obscure the most obvious findings. To
avoid such possibilities in the present work, the author will focus only on the fundamental
relationships that are the subject of this paper.
2. The Three Primary Types of Forces
To facilitate understanding of the mathematical treatment of force it is important to first
understand that there are three primary types of force. And that distinguishing between them can
be very confusing, yet is essential to proper use of the mathematics that defines them. For
example, in the analyses that follow, it will be seen that Newton’s second law of motion is both,
correct and incorrect in its present form depending on its usage. With that in mind, the three
primary types of forces are defined as follows:
a. Dual Frame Acceleration Force
A repulsive force that is applied between two masses causing acceleration of each mass
in inverse proportion to the amount of its mass relative to the other mass. If the primary mass is
vastly greater than the secondary mass, such as the Earth verses a particle being accelerated in a
particle accelerator, the primary mass is, for all practical purposes, not effected and the
secondary mass does all of the acceleration. Such dual frame forces can consist of mechanical
means, chemical means, electromagnetic means, etc., that are applied by way of exertion from
the primary mass to the secondary mass between the two masses. It is this type of force for
which Newton’s second law of motion is invalid in its present form due to the fact that such
forces have inherent speed limits that are absent in Newton’s second law.
b. Single Frame Acceleration Force
A repulsive force that is exerted against the same mass that contains the source of the
force. Unlike the dual frame force, this force does not require a second body in which to react to.
It is a reactionary force resulting from the expulsion of physical, chemical, electromagnetic, etc.,
entities contained within it. Assuming such entities are replenishable as the accelerating mass
travels through space, there is no limit to the speed at which the mass can be accelerated relative
to the frame at which it started along its path of travel. This is the acceleration frame force for
which Newton’s second law of motion is theoretically valid in its present form.
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c. Gravitational Force
An attractive force between two bodies of mass. This force has the reverse
characteristics of the single frame acceleration force. It is a force that is experienced by a body
that does not undergo accelerated motion. For example, a body at rest on the surface of the
Earth. This behavior is opposite to that of the single frame acceleration force yet is the very
behavior involved in the relativistic gravitational equivalence principle3. I.e., in a closed
container, an observer with no contact with the outside cannot distinguish between this
gravitational force and the single frame acceleration force previously given. The same holds true
for gravitational free fall and constant inertial motion or inertial rest state. Gravitational force is
not experienced by a body that does not resist it.
In pondering all of the possible ramifications of the three primary types of forces just
described it is easily seen how confusing it can be to distinguish between them. In that regard
the mathematical definitions are essential to an improved understanding of such forces. In
combination with the evidence, they help us to determine and predict the behavior being
investigated. Let us now examine and define the mathematics behind these three types of forces
and their relationships to Einstein’s energy equation and other principles of relativistic4 and
Newtonian physics.
3. Relativizing Newton’s Inertial Force Equation
We begin the mathematical examination and derivation process by immediately
modifying Newton’s second law of motion to make it agree with the dual frame acceleration
force previously described. The first step of the modification process is to convert the rest mass
defined as m0 in Einstein’s theory to its relativistic form defined as m in both, the E = mc2
equation and the newly revised F = ma equation. This is accomplished in the following equation
1 where m is the relativistic mass, m0 is the rest mass, and γ is the gamma symbol for the Lorentz
gamma factor
1
2 1
used in Einstein’s special theory of relativity. By simply using this form of m for the mass in
Newton’s inertial force equation,
3 we have made it compatible with Einstein’s energy equation.
4. Relating E = mc2 to the Revised F = ma Equation
To relate Einstein’s energy equation to the revised Newtonian inertial force equation we
begin by solving the revised force equation (3) for mass m. This gives
3
4
that by way of substitution into Einstein’s energy equation
5 gives
6
that can be simplified into its final form
7 of the revised energy equation. Solving the just derived equation for F gives
8 for the relativistic inertial force relationship to E = mc2.
5. Interpreting the New Force Behavior
To understand the real reason for the required modification to Newton’s inertial force
equation we need to analyze the full version of the new formula. For that version we simply take
equation (3) and through the process of substitution with equation (1) for mass m obtain
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which, by way of substitution with equation (2) for γ gives
1
10 1
for the complete equation. In analyzing the full version of the γ factor it is seen that for speed c
= 299792458 m/s and the speed of mass m0 having a value of v = 0 m/s, the γ factor = 1/1 or 1
and therefore has no effect on the force. For the other extreme where v = c the γ factor = 1/0 or
∞ and therefore mass m0 can never be accelerated to speed c because to do so would require an
infinite amount of force. Simply put, as v increases from 0 to c, the force F required for
continued acceleration increases from 0 to ∞ and therefore speed c can never be reached. Or,
stated another way, light speed c is a speed limit that can never be achieved by mass m0
regardless of how much force (short of infinite) is applied in the direction of acceleration. In that
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sense the new force equation is a general equation for inertial force since most methods of
applying force have speed limits that are insignificant with regard to light speed c. Only in the
case of particle accelerators and similar applications of force would the new force equation be
valid in its revised form as will be discussed later. But before proceeding there is a final issue to
be clarified.
In regard to all forms of the new force equation, whether given in the full form of
equation (10) or not, is the fact that acceleration rate a also involves speed v. That is, the
formula for constant acceleration is
11 where t is the time period during which the acceleration takes place. However, if we substitute
the right side of this equation in place of a in equation (10) we obtain
1
12 1
that simplifies into
13
1
1
14
1
1
15
1
giving
1
1
1
16 for the alternate form of the equation. This has no effect on the outcome of the equation and the
force F still increases from 0 to ∞ as v increases from 0 to c. The main point that the author is
making is that none of this behavior has anything to do with mass m0 increasing. It has to do
with a speed limit involved in the exertion of force relating to the method of application. And
the same is true of Einstein’s energy equation as will be covered next.
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6. Reinterpreting Einstein’s Energy Equation
As was the case with the force equation, to understand the true nature of Einstein’s
energy equation we need to analyze the full version of the formula. In that regard we take
equation (5) and through the process of substitution with equation (1) obtain
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into which we then substitute the right side of equation (2) for γ to obtain
1
18 1
that simplifies to
19
20
√
giving
√
21 for the complete version of the energy equation. And similar to the force analysis, it is seen that
as speed v increases from 0 to c, the effect on total energy (kinetic + internal energy of mass m0)
increases from 0 to ∞. Thus, there is an upper limit to the energy associate with mass m0. And,
as before, none of this has anything to do with mass m0 increasing. It has to do with the speed at
which mass m0 travels through space and the inability of the applied force to accelerate it to
speed limit c. This speed limit is inherent in the force method of application and has nothing to
do with an increase in the mass of the object being accelerated. This speed limit will be
discussed in more detail as we proceed.
At this point it is important to mention that this finding does not mean that mass m0
cannot be accelerated to speeds greater than speed c. It only means that mass m0 cannot be
accelerated to or beyond speed c relative to the frame of reference from which the force is being
applied. And, as previously mentioned, this limit involves the method of application and not the
mass that is being accelerated. I.e., in the case of particle accelerators the particles can never be
accelerated to or beyond speed c relative to the frame of the particle accelerator, i.e., Earth
frame. This limitation was covered in detail in an earlier work5 by this author published in
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March of 2007. With all of this understood we are then ready to discuss the speed limit factor in
a more generalized sense as will be accomplished next.
7. The Limited Validity of the γ Speed Limit Factor
The author makes no claim that the Lorentz gamma factor given in equation (2) is the
correct speed limit factor even with regard to particle accelerators and other applications
involving electromagnetic propagation. He simply acknowledges that, until someone can show
otherwise, based on the evidence it is the accepted factor for such applications. It is clear,
however, that it is not the correct factor for speed limits that do not approach the speed of light c.
I.e., speed limits associated with Newtonian mechanics. For example, planes, trains, and
automobiles. Yet, even in these cases it might only require a sight modification of the gamma
factor to make it compatible with these much lower speeds. That is, by simply taking the
previously give equation (10) for inertial force and redefining the speed limit as vL in place of
light speed c, we obtain
1
22 1
where vL can have any evidence determined value in the range of 0 ≤ vL < c. In other words, it is
reasonable to believe that the overall behavior of the previously defined Dual Frame
Acceleration Force of Section 2 is fundamentally the same for all types of such primary forces.
Alternatively, to what was just discussed, it is also possible that scientists may already
have devise alternative factors that precisely represent many of the speed limits in question and
could use them in place of the γ factor of the new force equation (9). It could be something as
simple as
1
23 1
for example, where ԑ is the speed limit factor, vL is the speed limit, and v is the speed reached by
mass m0 at that instant. By substitution into equation (9) we would then have
24 for the force associated with a specific area of Newtonian mechanics. Of course, the speed limit
factor could also be far more complicated than the present γ factor. Speculation on this issue,
however, outside of that already discussed is beyond the scope of this paper.
8. Validation of Newton’s Second Law of Motion
To validate the current nonrelativistic version of Newton’s second law of motion we have
to look no further than the mathematical definition of weight. The equation for weight is given
as
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25 where W is weight, and the symbol m for mass was replaced with m0 to make it compatible with
the mathematical convention used in this paper. The symbol g is for gravitational acceleration
and is defined by Newton’s equation for universal gravitation rearranged as
26 where F is gravitational force, m0 is the secondary mass, G is the gravitational constant, M is the
mass of the Earth, and R is the Earth’s radius. Since
27 as determined from equation (26) we then have
28
in relation to the weight equation (25) previously given. Both, the weight equation (25) and the
gravitational force equation (27) give the force as experienced in the gravitational frame defined
previously in Section 2 under the subheading, c. Gravitational Force. By way of comparison, it
is readily seen that Newton’s gravitational force equation (27) is fundamentally the same as his
unrevised inertial force equation given here as equation (29)
29 where m0 replaces m for consistency with the mathematical convention used. It is this
mathematical relationship between gravitational force and inertial force that provides the basis
for Einstein’s gravitational equivalence principle and thus validates this nonrelativistic form of
the inertial force equation along with the relativistic version given previously as equations (3),
(9), and (10).
9. The Significance of the Weight-Force Relationship
The real significance of the weight-force relationship just covered involves what it tells
us about mass. When the gravitational force increases due to an increase in the gravitational
acceleration rate, we don’t conclude that the effected mass m0 has increased. We attribute the
increase in force to an increase in weight. Why then should we attribute an increase in inertial
force to an increase in mass when inertial acceleration increases? I.e., an object transported to
the surface of a more massive body in space is said to have a greater weight than it had on the
home body due to the increase in gravitational force even though its own mass has not changed.
This would be true even if the more massive body was a black hole in which case the
gravitational force on its surface is greater than that experienced by the same object being
accelerated to the speed of light relative to its original Earth frame. Yet, a particle accelerated to
near the speed of light in a particle accelerator is said to have an increase in mass and not an
increase in weight. This treatment of gravitational versus inertial force is in violation of the
equivalence principle of Einstein’s theory of relativity as will be shown next.
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10. Proof that Equivalence Principle is Non-relativistic
We can express the just discussed gravitational-inertial relationship mathematically by
equating the previously derived gravitational force equation (27) to the previously derived
relativistic inertial force equation (9). This gives
30 for the Newtonian gravitational force and
31 for the relativistic inertial force where the subscripts were added to distinguish between the two
different types of forces. On the condition that
32
we obtain
33
giving
34 for the gravitational force, and
35 for the relativistic inertial force in relation to each other. It is clear from this comparison that
Newtonian gravitational force and relativistic inertial force are not equal. On the other hand, if
we take the Newtonian inertial force equation (29) (with the addition of the identifying Ni
subscript) and equate it with the Newtonian gravitational force equation (30) in the same manner,
we obtain
36
giving
37 for the true gravitational-inertial equivalence principle. But as can be seen, this involves only
Newtonian principles and not the relativistic principles of Einstein’s theory. That is, in referring
back to the three primary types of forces in Section 2, it is seen that Newtonian inertial force is a
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single frame acceleration force. And as stated in the gravitational force description, these two
forces are indistinguishable from each other when experienced in a closed container.
11. Relationship of Newtonian to Relativistic Inertial Force
The relationship of Newtonian inertial force to relativistic inertial force is rather straight
forward and best explained mathematically. Given the mathematical relationship of gravitational
force to relativistic inertial force of equation (34) and the gravitational-inertial equivalence of
equation (37) by way of substitution we obtain
38
where FNi is the Newtonian inertial force and Fri is the relativistic inertial force. Applying the
full version of gamma then gives
1
39
1
that simplifies to
1
40 where FNi is the single frame acceleration force experienced in the acceleration frame accelerated
by an external, dual frame acceleration force, Fri. By analyzing the Lorentz factor it can be seen
that as the speed v from acceleration increases from 0 to c, the force FNi experienced in the
acceleration frame decreases from Fri to 0. This is exactly what would be expected under these
conditions. No matter how powerful the applied external force Fri is, it has reached its speed
limit at the end of this period and thus the resulting acceleration of the object being accelerated
has reduced to 0. Such object would now be traveling at close to light speed c relative to the
inertial frame of the applied force Fri.
12. Final Considerations
Before proceeding to the conclusion of this paper there are a few final considerations that
bear mention. They are given next in alphabetic order.
A. The first involves the gravitational equivalence principle. It is impossible to escape
the fact that there is a second aspect to the equivalence principle, one regarding gravitational free
fall and its relationship to inertial motion. In the same manner that the gravitational force on a
body at rest on a gravitational mass cannot be distinguished from the force of inertial
acceleration in a closed container, the absence of such force experience during gravitational free
fall makes such accelerated motion indistinguishable from uniform inertial motion or an inertial
state of rest *. It is important to note that in both cases (object at rest on surface of massive body
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vs accelerating object in space) and (object at rest or uniform motion in space vs object in free
fall toward massive body) a state of rest and the corresponding state of motion are
indistinguishable from each other in a closed container. (* The author has shown repeatedly in
his work6 that uniform motion and a state of rest are the same thing.)
Also of importance is the fact that the equivalence principle is meant in the general sense,
since there are countless specifics where it can be shown that, even in a closed container, there
are ways to distinguish between the two different forms of force and motion. However, for every
argument against the equivalence principle there is a counter argument in support of it. And to
cover the subject in any kind of detail would only serve to clutter the findings presented in this
paper.
B. The next issue that needs addressing involves an inherent limitation regarding the
mathematical definition of dual frame acceleration force. Such force is defined mathematically
as the physical cause of acceleration. This means that any increase in its application after the
speed limit is reached has no mathematical definition in current physics. I.e., when the speed
limit is reached the acceleration reduces to 0 and the acted upon object continues to move in
uniform motion even if the amount of force has been tripled after the speed limit was reached.
We can’t deny that this increase is in effect even though it has no effect on the motion of the
object on which it is applied and therefore does not show up as an increase in the value of F.
This increase in force is not accounted for in Einstein’s mass increase definition either. If it is
not force, then what is it?
C. In reference to the energy and force equations that use light speed c as the speed limit;
if at some future time the speed limit is exceeded, it does not mean the respective energy or force
is infinite. It only means that c was not the speed limit after all and the speed limit has to be
increased to represent the new limit.
D. It should be understood that the speed limit is actually more than a speed limit. It is a
speed that cannot be achieved at all. It is actually an off limit speed.
13. Conclusion
The author makes no pretense that the subject matter presented in this paper is easy to
comprehend. To the contrary, what is shown was so difficult to work out that the author had to
modify the title of the paper several times during its creation. Right out to the moment of
completion, changes and updates had to be incorporated supported by mathematical analyses
conducted in an advanced computer application. And none of this had anything to do with the
original intention of showing that the real reason for the relativistic behavior of inertial force and
Einstein’s energy equation involves a speed limit and not a mass increase as given in Einstein’s
theory. It all had to do with the difficulty of presenting the findings in an easy and intuitive
manner for the purpose of clarity and complete understanding. Hopefully the author has
succeeded and the reader was able to fully comprehend the material presented.
REFERENCES
1
The energy equation E = mc2 first appeared in a paper by Max Planck, “The Principle of Relativity and the
Fundamental Equations of Mechanics,” Verhandlungen der Deutschen Physikalischen Gesellschaft 4 (1906):136141. With regard to Einstein, the energy equation is first mentioned in the paper, Does the inertia of a body depend
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upon its energy-content? (1905), It is unclear when the equation was first given in the form E = mc2, Einstein’s first
use of the equation was in, “On the Inertia of Energy Required by the Relativity Principle,” Annalen der Physik 23
(1907):371-384.
2
Joseph A. Rybczyk, The Relationship between E = Mc2 and F = ma, (2007), Available at www.mrelativity.net
3
Similar ideas had been expressed by Bishop Berkley in 1721. Einstein, The Meaning of Relativity, 58.
4
Principles involving Einstein’s Theory of Relativity, Most notably as given in, Albert Einstein, the Special Theory
of Relativity, originally published under the title, On the Electrodynamics of Moving Bodies, Annalen der Physik,
17, (1905)
5
Joseph A. Rybczyk, Breaking the Light Speed Barrier, (2007), Available at www.mrelativity.net
6
All of the author’s papers are available at the Millennium Relativity Website at http://mrelativity.net/
Relationship of E = mc2 to F = ma and Gravity
Copyright © 2016
Joseph A. Rybczyk
All rights reserved
including the right of reproduction
in whole or in part in any form
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